#7092821

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1383 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,26)(19,24,28,40)(20,25,33,41)(21,23,34,27)(29,39,32,42)(30,36,44,38)(31,43,35,37)(45,47,46,48);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,14)(13,16)(18,23)(19,28)(20,31)(21,34)(24,38)(25,41)(26,27)(29,45)(32,46)(33,35)(36,40)(47,48);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,32)(21,30)(22,28)(23,36)(25,42)(26,40)(27,38)(29,33)(31,45)(34,44)(35,46)(37,48)(39,41)(43,47);(11,16)(13,15)(17,20,22,33)(18,25,26,41)(19,29,28,32)(21,35,34,31)(23,37,27,43)(24,39,40,42)(30,45,44,46)(36,47,38,48);(1,4,5,3)(2,7,8,6)(10,14)(13,15)(18,26)(19,28)(23,27)(25,41)(29,32)(30,44)(37,43)(45,46)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,d^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=ehi,f^b=fi,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7000017);